A002973 -id:A002973 - cp's OEIS frontend

A002144 Pythagorean primes: primes of the form 4*k + 1.

5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 313, 317, 337, 349, 353, 373, 389, 397, 401, 409, 421, 433, 449, 457, 461, 509, 521, 541, 557, 569, 577, 593, 601, 613, 617

Offset: 1

N. J. A. Sloane

Rational primes that decompose in the field Q(sqrt(-1)). - N. J. A. Sloane, Dec 25 2017
These are the prime terms of A009003.
-1 is a quadratic residue mod a prime p if and only if p is in this sequence.
Sin(a(n)*Pi/2) = 1 with Pi = 3.1415..., see A070750. - Reinhard Zumkeller, May 04 2002
If at least one of the odd primes p, q belongs to the sequence, then either both or neither of the congruences x^2 = p (mod q), x^2 = q (mod p) are solvable, according to Gauss reciprocity law. - Lekraj Beedassy, Jul 17 2003
Odd primes such that binomial(p-1, (p-1)/2) == 1 (mod p). - Benoit Cloitre, Feb 07 2004
Primes that are the hypotenuse of a right triangle with integer sides. The Pythagorean triple is {A002365(n), A002366(n), a(n)}.
Also, primes of the form a^k + b^k, k > 1. - Amarnath Murthy, Nov 17 2003
The square of a(n) is the average of two other squares. This fact gives rise to a class of monic polynomials x^2 + bx + c with b = a(n) that will factor over the integers regardless of the sign of c. See A114200. - Owen Mertens (owenmertens(AT)missouristate.edu), Nov 16 2005
Also such primes p that the last digit is always 1 for the Nexus numbers of form n^p - (n-1)^p. - Alexander Adamchuk, Aug 10 2006
The set of Pythagorean primes is a proper subset of the set of positive fundamental discriminants (A003658). - Paul Muljadi, Mar 28 2008
A079260(a(n)) = 1; complement of A137409. - Reinhard Zumkeller, Oct 11 2008
From Artur Jasinski, Dec 10 2008: (Start)
If we take 4 numbers: 1, A002314(n), A152676(n), A152680(n) then multiplication table modulo a(n) is isomorphic to the Latin square:
   1 2 3 4
   2 4 1 3
   3 1 4 2
   4 3 2 1
and isomorphic to the multiplication table of {1, i, -i, -1} where i is sqrt(-1), A152680(n) is isomorphic to -1, A002314(n) with i or -i and A152676(n) vice versa -i or i. 1, A002314(n), A152676(n), A152680(n) are subfield of Galois field [a(n)]. (End)
Primes p such that the arithmetic mean of divisors of p^3 is an integer. There are 2 sequences of such primes: this one and A002145. - Ctibor O. Zizka, Oct 20 2009
Equivalently, the primes p for which the smallest extension of F_p containing the square roots of unity (necessarily F_p) contains the 4th roots of unity. In this respect, the n = 2 case of a family of sequences: see n=3 (A129805) and n=5 (A172469). - Katherine E. Stange, Feb 03 2010
Subsequence of A007969. - Reinhard Zumkeller, Jun 18 2011
A151763(a(n)) = 1.
k^k - 1 is divisible by 4*k + 1 if 4*k + 1 is a prime (see Dickson reference). - Gary Detlefs, May 22 2013
Not only are the squares of these primes the sum of two nonzero squares, but the primes themselves are also. 2 is the only prime equal to the sum of two nonzero squares and whose square is not. 2 is therefore not a Pythagorean prime. - Jean-Christophe Hervé, Nov 10 2013
The statement that these primes are the sum of two nonzero squares follows from Fermat's theorem on the sum of two squares. - Jerzy R Borysowicz, Jan 02 2019
The decompositions of the prime and its square into two nonzero squares are unique. - Jean-Christophe Hervé, Nov 11 2013. See the Dickson reference, Vol. II, (B) on p. 227. - Wolfdieter Lang, Jan 13 2015
p^e for p prime of the form 4*k+1 and e >= 1 is the sum of 2 nonzero squares. - Jon Perry, Nov 23 2014
Primes p such that the area of the isosceles triangle of sides (p, p, q) for some integer q is an integer. - Michel Lagneau, Dec 31 2014
This is the set of all primes that are the average of two squares. - Richard R. Forberg, Mar 01 2015
Numbers k such that ((k-3)!!)^2 == -1 (mod k). - Thomas Ordowski, Jul 28 2016
This is a subsequence of primes of A004431 and also of A016813. - Bernard Schott, Apr 30 2022
In addition to the comment from Jean-Christophe Hervé, Nov 10 2013: All powers as well as the products of any of these primes are the sum of two nonzero squares. They are terms of A001481, which is closed under multiplication. - Klaus Purath, Nov 19 2023

			The following table shows the relationship between several closely related sequences:
Here p = A002144 = primes == 1 (mod 4), p = a^2+b^2 with a < b;
a = A002331, b = A002330, t_1 = ab/2 = A070151;
p^2 = c^2 + d^2 with c < d; c = A002366, d = A002365,
t_2 = 2ab = A145046, t_3 = b^2 - a^2 = A070079,
with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).
  ---------------------------------
   p  a  b  t_1  c   d t_2 t_3  t_4
  ---------------------------------
   5  1  2   1   3   4   4   3    6
  13  2  3   3   5  12  12   5   30
  17  1  4   2   8  15   8  15   60
  29  2  5   5  20  21  20  21  210
  37  1  6   3  12  35  12  35  210
  41  4  5  10   9  40  40   9  180
  53  2  7   7  28  45  28  45  630
  ...
a(7) = 53 = A002972(7)^2 + (2*A002973(7))^2 = 7^2 + (2*1)^2 = 49 + 4, and this is the only way. - _Wolfdieter Lang_, Jan 13 2015

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.
L. E. Dickson, "History of the Theory of Numbers", Chelsea Publishing Company, 1919, Vol I, page 386
L. E. Dickson, History of the Theory of Numbers, Carnegie Institution, Publ. No. 256, Vol. II, Washington D.C., 1920, p. 227.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 132.
M. du Sautoy, The Music of the Primes, Fourth Estate / HarperCollins, 2003; see p. 76.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 241, 243.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 90.

Zak Seidov, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972.
C. Banderier, Calcul de (-1/p).
J. Butcher, Mathematical Miniature 8: The Quadratic Residue Theorem, NZMS Newsletter, No. 75, April 1999.
Hing Lun Chan, Windmills of the minds: an algorithm for Fermat's Two Squares Theorem, arXiv:2112.02556 [cs.LO], 2021.
R. Chapman, Quadratic reciprocity.
A. David Christopher, A partition-theoretic proof of Fermat's Two Squares Theorem, Discrete Mathematics, Volume 339, Issue 4, 6 April 2016, Pages 1410-1411.
J. E. Ewell, A Simple Proof of Fermat's Two-Square Theorem, The American Mathematical Monthly, Vol. 90, No. 9 (Nov., 1983), pp. 635-637.
Bernard Frénicle de Bessy, Traité des triangles rectangles en nombres : dans lequel plusieurs belles propriétés de ces triangles sont démontrées par de nouveaux principes, Michalet, Paris (1676) pp. 0-116; see p. 44, Consequence II.
Bernard Frénicle de Bessy, Méthode pour trouver la solution des problèmes par les exclusions. Abrégé des combinaisons. Des Quarrez magiques, in "Divers ouvrages de mathématiques et de physique, par MM. de l'Académie royale des sciences", (1693) "Troisième exemple", pp. 17-26, see in particular p. 25.
A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
D. & C. Hazzlewood, Quadratic Reciprocity.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Lucas Lacasa, Bartolome Luque, Ignacio Gómez, and Octavio Miramontes, On a Dynamical Approach to Some Prime Number Sequences, Entropy 20.2 (2018): 131, also arXiv:1802.08349 [math.NT], 2018.
R. C. Laubenbacher and D. J. Pengelley, Eisenstein's Misunderstood Geometric Proof of the Quadratic Reciprocity Theorem, In: Anderson M, Katz V, Wilson R, eds. Who Gave You the Epsilon?: And Other Tales of Mathematical History. Spectrum. Mathematical Association of America; 2009:309-312.
R. C. Laubenbacher and D. J. Pengelley, Gauss, Eisenstein and the 'third' proof of the Quadratic Reciprocity Theorem, The Mathematical Intelligencer 16, 67-72 (1994).
K. Matthews, Serret's algorithm based Server.
Jorma K. Merikoski, Pentti Haukkanen, and Timo Tossavainen, The congruence x^n = -a^n (mod m): Solvability and related OEIS sequences, Notes. Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 516-529. See p. 521.
Carlos Rivera, Puzzle 968. Another property of primes 4m+1, The Prime Puzzles & Problems Connection.
D. Shanks, Review of "K. E. Kloss et al., Class number of primes of the form 4n+1", Math. Comp., 23 (1969), 213-214. [Annotated scanned preprint of review]
S. A. Shirali, A family portrait of primes-a case study in discrimination, Math. Mag. Vol. 70, No. 4 (Oct., 1997), pp. 263-272.
Rosemary Sullivan and Neil Watling, Independent divisibility pairs on the set of integers from 1 to n, INTEGERS 13 (2013) #A65.
Eric Weisstein's World of Mathematics, Wilson's Theorem.
Eric Weisstein's World of Mathematics, Pythagorean Triples.
Wikipedia, Quadratic reciprocity
Wolfram Research, The Gauss Reciprocity Law.
G. Xiao, Two squares.
D. Zagier, A One-Sentence Proof That Every Prime p == 1 (mod 4) Is a Sum of Two Squares, Am. Math. Monthly, Vol. 97, No. 2 (Feb 1990), p. 144. [From _Wolfdieter Lang_, Jan 17 2015 (thanks to Charles Nash)]
Index to sequences related to decomposition of primes in quadratic fields.

Cf. A002145, A002314, A002476, A002972, A002973, A003658, A004431, A007519, A010051, A016813, A076339, A094407.
Cf. A114200, A133870, A142925, A152676, A152680, A173330, A173331, A208177, A208178, A334912.
Cf. A004613 (multiplicative closure).
Apart from initial term, same as A002313.
For values of n see A005098.
Primes in A020668.

a002144 n = a002144_list !! (n-1)
a002144_list = filter ((== 1) . a010051) [1,5..]
-- Reinhard Zumkeller, Mar 06 2012, Feb 22 2011

[a: n in [0..200] | IsPrime(a) where a is 4*n + 1 ]; // Vincenzo Librandi, Nov 23 2014

a := []; for n from 1 to 500 do if isprime(4*n+1) then a := [op(a),4*n+1]; fi; od: A002144 := n->a[n];
# alternative
A002144 := proc(n)
    option remember ;
    local a;
    if n = 1 then
        5;
    else
        for a from procname(n-1)+4 by 4 do
            if isprime(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A002144(n),n=1..100) ; # R. J. Mathar, Jan 31 2024

Select[4*Range[140] + 1, PrimeQ[ # ] &] (* Stefan Steinerberger, Apr 16 2006 *)
Select[Prime[Range[150]],Mod[#,4]==1&] (* Harvey P. Dale, Jan 28 2021 *)

PARI
```
select(p->p%4==1,primes(1000))
```

A002144_next(p=A2144[#A2144])={until(isprime(p+=4),);p} /* NB: p must be of the form 4k+1. Beyond primelimit, this is *much* faster than forprime(p=...,, p%4==1 && return(p)). */
A2144=List(5); A002144(n)={while(#A2144A002144_next())); A2144[n]}
\\ M. F. Hasler, Jul 06 2024

from sympy import prime
A002144 = [n for n in (prime(x) for x in range(1,10**3)) if not (n-1) % 4]
# Chai Wah Wu, Sep 01 2014

from sympy import isprime
print(list(filter(isprime, range(1, 618, 4)))) # Michael S. Branicky, May 13 2021

def A002144_list(n): # returns all Pythagorean primes <= n
    return [x for x in prime_range(5,n+1) if x % 4 == 1]
A002144_list(617) # Peter Luschny, Sep 12 2012

Odd primes of form x^2 + y^2, (x=A002331, y=A002330, with x < y) or of form u^2 + 4*v^2, (u = A002972, v = A002973, with u odd). - Lekraj Beedassy, Jul 16 2004
p^2 - 1 = 12*Sum_{i = 0..floor(p/4)} floor(sqrt(i*p)) where p = a(n) = 4*n + 1. [Shirali]
a(n) = A000290(A002972(n)) + A000290(2*A002973(n)) = A000290(A002331(n+1)) + A000290(A002330(n+1)). - Reinhard Zumkeller, Feb 16 2010
a(n) = A002972(n)^2 + (2*A002973(n))^2, n >= 1. See the Jean-Christophe Hervé Nov 11 2013 comment. - Wolfdieter Lang, Jan 13 2015
a(n) = 4*A005098(n) + 1. - Zak Seidov, Sep 16 2018
From Vaclav Kotesovec, Apr 30 2020: (Start)
Product_{k>=1} (1 - 1/a(k)^2) = A088539.
Product_{k>=1} (1 + 1/a(k)^2) = A243380.
Product_{k>=1} (1 - 1/a(k)^3) = A334425.
Product_{k>=1} (1 + 1/a(k)^3) = A334424.
Product_{k>=1} (1 - 1/a(k)^4) = A334446.
Product_{k>=1} (1 + 1/a(k)^4) = A334445.
Product_{k>=1} (1 - 1/a(k)^5) = A334450.
Product_{k>=1} (1 + 1/a(k)^5) = A334449. (End)
From Vaclav Kotesovec, May 05 2020: (Start)
Product_{k>=1} (1 + 1/A002145(k)) / (1 + 1/a(k)) = Pi/(4*A064533^2) = 1.3447728438248695625516649942427635670667319092323632111110962...
Product_{k>=1} (1 - 1/A002145(k)) / (1 - 1/a(k)) = Pi/(8*A064533^2) = 0.6723864219124347812758324971213817835333659546161816055555481... (End)
Sum_{k >= 1} 1/a(k)^s = (1/2) * Sum_{n >= 1 odd numbers} moebius(n) * log((2*n*s)! * zeta(n*s) * abs(EulerE(n*s - 1)) / (Pi^(n*s) * 2^(2*n*s) * BernoulliB(2*n*s) * (2^(n*s) + 1) * (n*s - 1)!))/n, s >= 3 odd number. - Dimitris Valianatos, May 21 2020
Legendre symbol (-1, a(n)) = +1, for n >= 1. - Wolfdieter Lang, Mar 03 2021

A002330 Value of y in the solution to p = x^2 + y^2, x <= y, with prime p = A002313(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 5, 7, 6, 8, 8, 9, 10, 10, 8, 11, 10, 11, 13, 10, 12, 14, 15, 13, 15, 16, 13, 14, 16, 17, 13, 14, 16, 18, 17, 18, 17, 19, 20, 20, 15, 17, 20, 21, 19, 22, 20, 21, 19, 20, 24, 23, 24, 18, 19, 25, 22, 25, 23, 26, 26, 22, 27, 26, 20, 25, 22, 26, 28, 25

Offset: 1

N. J. A. Sloane

nonn

			The following table shows the relationship between several closely related sequences:
Here p = A002144 = primes == 1 (mod 4), p = a^2+b^2 with a < b;
a = A002331, b = A002330, t_1 = ab/2 = A070151;
p^2 = c^2+d^2 with c < d; c = A002366, d = A002365,
t_2 = 2ab = A145046, t_3 = b^2-a^2 = A070079,
with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).
---------------------------------
.p..a..b..t_1..c...d.t_2.t_3..t_4
---------------------------------
.5..1..2...1...3...4...4...3....6
13..2..3...3...5..12..12...5...30
17..1..4...2...8..15...8..15...60
29..2..5...5..20..21..20..21..210
37..1..6...3..12..35..12..35..210
41..4..5..10...9..40..40...9..180
53..2..7...7..28..45..28..45..630
.................................

A. J. C. Cunningham, Quadratic Partitions. Hodgson, London, 1904, p. 1.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
John Brillhart, Note on representing a prime as a sum of two squares, Math. Comp. 26 (1972), pp. 1011-1013.
A. J. C. Cunningham, Quadratic Partitions, Hodgson, London, 1904. [Annotated scans of selected pages]
K. Matthews, Serret's algorithm Server.
J. Todd, A problem on arc tangent relations, Amer. Math. Monthly, 56 (1949), 517-528.
Eric Weisstein's World of Mathematics, Fermat's 4n Plus 1 Theorem.

Cf. A002331, A002313, A002144.

a := []; for x from 0 to 50 do for y from x to 50 do p := x^2+y^2; if isprime(p) then a := [op(a),[p,x,y]]; fi; od: od: writeto(trans); for i from 1 to 158 do lprint(a[i]); od: # then sort the triples in "trans"

Flatten[#, 1]&[Table[PowersRepresentations[Prime[k], 2, 2], {k, 1, 142}]][[All, 2]] (* Jean-François Alcover, Jul 05 2011 *)

f(p)=my(s=lift(sqrt(Mod(-1,p))),x=p,t);if(s>p/2,s=p-s); while(s^2>p, t=s;s=x%s;x=t);s
forprime(p=2,1e3,if(p%4-3,print1(f(p)", "))) \\ Charles R Greathouse IV, Apr 24 2012

do(p)=qfbsolve(Qfb(1,0,1),p)[1]
forprime(p=2,1e3,if(p%4-3,print1(do(p)", "))) \\ Charles R Greathouse IV, Sep 26 2013

print1(1); forprimestep(p=5,1e3,4, print1(", "qfbcornacchia(1,p)[1])) \\ Charles R Greathouse IV, Sep 15 2021

a(n) = A096029(n) + A096030(n) + 1, for n>1. - Lekraj Beedassy, Jul 21 2004

a(n+1) = Max(A002972(n), 2*A002973(n)). - Reinhard Zumkeller, Feb 16 2010

A005098 Numbers k such that 4k + 1 is prime.

Original entry on oeis.org

1, 3, 4, 7, 9, 10, 13, 15, 18, 22, 24, 25, 27, 28, 34, 37, 39, 43, 45, 48, 49, 57, 58, 60, 64, 67, 69, 70, 73, 78, 79, 84, 87, 88, 93, 97, 99, 100, 102, 105, 108, 112, 114, 115, 127, 130, 135, 139, 142, 144, 148, 150, 153, 154, 160, 163, 165, 168, 169, 175, 177, 183

Offset: 1

N. J. A. Sloane

Sum of i-th and j-th triangular numbers, where i=A096029(n), j=A096030(n); i.e., a(n) = A000217(A096029(n)) + A000217(A096030(n)). - Lekraj Beedassy, Jun 16 2004
For every k in the sequence, there is exactly 1 square number that can be subtracted to leave a pronic (A002378). E.g., 27 - 25 = 2, 99 - 9 = 90. - Jon Perry, Nov 06 2010
See A208295 for details concerning the preceding Jon Perry comment. - Wolfdieter Lang, Mar 29 2012
a(k) appears in the o.g.f. for floor(A002144(k)*j^2/4), j >= 0, for k >= 1: x*(a(k)*(1 + x^2) + b(k)*x)/((1 - x)^3*(1 + x)), together with b(k) = (A002144(k) + 1)/2 = A119681(k). - Wolfdieter Lang, Aug 07 2013

Zak Seidov, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Eric Weisstein's World of Mathematics, Wilson's Theorem.

See A002144 for the actual primes.

Cf. A002972, A002973, A173331, A173330, A023212.

a005098 = (`div` 4) . (subtract 1) . a002144
-- Reinhard Zumkeller, Mar 17 2013

[k: k in [0..10000] | IsPrime(4*k+1)] // Vincenzo Librandi, Nov 18 2010

a := []; for k from 1 to 500 do if isprime(4*k+1) then a := [op(a), k]; fi; od: A005098 := k->a[k];

Select[Range[200], PrimeQ[4# + 1] &] (* Harvey P. Dale, Apr 20 2011 *)

is(k)=isprime(4*k+1) \\ Charles R Greathouse IV, Nov 20 2012

a(n) = (A002144(n)-1)/4.

More terms from Ray Chandler, Jun 26 2004

Edited by Charles R Greathouse IV, Mar 17 2010

A002331 Values of x in the solution to p = x^2 + y^2, x <= y, with prime p = A002313(n).

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 4, 2, 5, 3, 5, 4, 1, 3, 7, 4, 7, 6, 2, 9, 7, 1, 2, 8, 4, 1, 10, 9, 5, 2, 12, 11, 9, 5, 8, 7, 10, 6, 1, 3, 14, 12, 7, 4, 10, 5, 11, 10, 14, 13, 1, 8, 5, 17, 16, 4, 13, 6, 12, 1, 5, 15, 2, 9, 19, 12, 17, 11, 5, 14, 10, 18, 4, 6, 16, 20, 19, 10, 13, 4, 6, 15, 22, 11, 3, 5

Offset: 1

N. J. A. Sloane

nonn

			The following table shows the relationship between several closely related sequences:
Here p = A002144 = primes == 1 (mod 4), p = a^2+b^2 with a < b;
a = A002331, b = A002330, t_1 = ab/2 = A070151;
p^2 = c^2+d^2 with c < d; c = A002366, d = A002365,
t_2 = 2ab = A145046, t_3 = b^2-a^2 = A070079,
with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).
---------------------------------
.p..a..b..t_1..c...d.t_2.t_3..t_4
---------------------------------
.5..1..2...1...3...4...4...3....6
13..2..3...3...5..12..12...5...30
17..1..4...2...8..15...8..15...60
29..2..5...5..20..21..20..21..210
37..1..6...3..12..35..12..35..210
41..4..5..10...9..40..40...9..180
53..2..7...7..28..45..28..45..630
.................................

A. J. C. Cunningham, Quadratic Partitions. Hodgson, London, 1904, p. 1.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
A. T. Benjamin and D. Zeilberger, Pythagorean primes and palindromic continued fractionsINTEGERS 5(1) (2005) #A30
John Brillhart, Note on representing a prime as a sum of two squares, Math. Comp. 26 (1972), pp. 1011-1013.
A. J. C. Cunningham, Quadratic Partitions, Hodgson, London, 1904. [Annotated scans of selected pages]
K. Matthews, Serret's algorithm Server.
J. Todd, A problem on arc tangent relations, Amer. Math. Monthly, 56 (1949), 517-528.
Eric Weisstein's World of Mathematics, Fermat's 4n Plus 1 Theorem.

Cf. A002330, A002313, A002144, A027862 (locates y=x+1).

See A002330 for Maple program.
# alternative
A002331 := proc(n)
    A363051(A002313(n)) ;
end proc:
seq(A002331(n),n=1..100) ; # R. J. Mathar, Feb 01 2024

pmax = 1000; x[p_] := Module[{x, y}, x /. ToRules[Reduce[0 <= x <= y && x^2 + y^2 == p, {x, y}, Integers]]]; For[n=1; p=2, pJean-François Alcover, Feb 26 2016 *)

f(p)=my(s=lift(sqrt(Mod(-1,p))),x=p,t);if(s>p/2,s=p-s); while(s^2>p,t=s;s=x%s;x=t);s
forprime(p=2,1e3,if(p%4-3,print1(sqrtint(p-f(p)^2)", ")))
\\ Charles R Greathouse IV, Apr 24 2012

do(p)=qfbsolve(Qfb(1,0,1),p)[2]
forprime(p=2,1e3,if(p%4-3,print1(do(p)", "))) \\ Charles R Greathouse IV, Sep 26 2013

a(n) = A096029(n) - A096030(n) for n > 1. - Lekraj Beedassy, Jul 16 2004
a(n+1) = Min(A002972(n), 2*A002973(n)). - Reinhard Zumkeller, Feb 16 2010
a(n) = A363051(A002313(n)). - R. J. Mathar, Jan 31 2024

A002972 a(n) is the odd member of {x,y}, where x^2 + y^2 is the n-th prime of the form 4i+1.

Original entry on oeis.org

1, 3, 1, 5, 1, 5, 7, 5, 3, 5, 9, 1, 3, 7, 11, 7, 11, 13, 9, 7, 1, 15, 13, 15, 1, 13, 9, 5, 17, 13, 11, 9, 5, 17, 7, 17, 19, 1, 3, 15, 17, 7, 21, 19, 5, 11, 21, 19, 13, 1, 23, 5, 17, 19, 25, 13, 25, 23, 1, 5, 15, 27, 9, 19, 25, 17, 11, 5, 25, 27, 23, 29, 29, 25, 23, 19, 29, 13, 31, 31

Offset: 1

N. J. A. Sloane

nonn

It appears that the terms in this sequence are the absolute values of the terms in A046730. - Gerry Myerson, Dec 02 2010

"the n-th prime of the form 4i+1" is A005098(n). - Rainer Rosenthal, Aug 24 2022

			The 2nd prime of the form 4i+1 is 13 = 2^2 + 3^2, so a(2)=3.

E. Kogbetliantz and A. Krikorian, Handbook of First Complex Prime Numbers, Gordon and Breach, NY, 1971, p. 243.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Rainer Rosenthal, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
S. R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
E. Kogbetliantz and A. Krikorian, Handbook of First Complex Prime Numbers, Gordon and Breach, NY, 1971. [Annotated scans of a few pages]
Stan Wagon, Editor's Corner: The Euclidean Algorithm Strikes Again, The American Mathematical Monthly, vol. 97, no. 2, 1990, pp. 125-29. [Description of efficient decomposition algorithm implemented in PARI program]

Cf. A002144, A002973, A005098, A261858.

pmax = 1000; odd[p_] := Module[{k, m}, 2m+1 /. ToRules[Reduce[k>0 && m >= 0 && (2k)^2 + (2m+1)^2 == p, {k, m}, Integers]]]; For[n=1; p=5, pJean-François Alcover, Feb 26 2016 *)

decomp2sq(p) = {my (m=(p-1)/4, r, x, limit=ceil(sqrt(p))); if (p>4 && denominator(m)==1, forprime (c=2,oo, if (!issquare(Mod(c,p)), r=c; break)); x=lift (Mod(r,p)^m); until (px%2,decomp2sq(p))[1],", "))) \\ Hugo Pfoertner, Aug 27 2022

a(n) = Min(A173330(n), A002144(n) - A173330(n)). - Reinhard Zumkeller, Feb 16 2010
a(n)^2 + 4*A002973(n)^2 = A002144(n); A002331(n+1) = Min(a(n),2*A002973(n)) and A002330(n+1) = Max(a(n),2*A002973(n)). - Reinhard Zumkeller, Feb 16 2010
(a(n) - 1)/2 = A208295(n), n >= 1. - Wolfdieter Lang, Mar 03 2012
a(A267858(k)) == 1 (mod 4), k >= 1. - Wolfdieter Lang, Feb 18 2016

Better description from Jud McCranie, Mar 05 2003

A173331 Second of two intermediate sequences for integral solution of A002144(n)=x^2+y^2.

Original entry on oeis.org

2, 2, 13, 2, 31, 4, 2, 55, 8, 81, 4, 91, 99, 105, 133, 10, 6, 2, 10, 181, 183, 227, 8, 237, 16, 10, 14, 265, 2, 301, 303, 16, 18, 8, 355, 379, 6, 381, 389, 14, 421, 429, 453, 451, 487, 20, 531, 543, 20, 24, 585, 24, 18, 16, 637, 631, 655, 12, 651, 675, 22, 731, 26, 741, 757

Offset: 1

Reinhard Zumkeller, Feb 16 2010

nonn

a(n) = A173330(n)*A010050(A005098(n)) mod A002144(n);

A002973(n) = MIN(a(n), A002144(n) - a(n)) / 2.

			n=7: A002144(7) = 53 = 4*13 + 1,
a(7) = A173330(7) * 26! mod 53 = 7*403291461126605635584000000 mod 53 = 2,
A002973(7) = MIN(2, 53 - 2) / 2 = 1;
n=8: A002144(8) = 61 = 4*15 + 1,
a(8) = A173330(8) * 30! mod 61 = 5*265252859812191058636308480000000 mod 61 = 55,
A002973(8) = MIN(55, 61 - 55) / 2 = 3.

H. Davenport, The Higher Arithmetic (Cambridge University Press 7th ed., 1999), ch. V.3, p.122.

Cf. A123072, A000142.

a(n) = ((2k)! / 2(k!))^2 mod p, where p = 4*k+1 = A002144(n).

A278720 The p-defect p - N(p) of the congruence y^2 == x^3 + 4*x (mod p) for primes p, where N(p) is the number of solutions given by A276730.

Original entry on oeis.org

0, 0, -2, 0, 0, 6, 2, 0, 0, -10, 0, -2, 10, 0, 0, 14, 0, -10, 0, 0, -6, 0, 0, 10, 18, -2, 0, 0, 6, -14, 0, 0, -22, 0, 14, 0, 22, 0, 0, -26, 0, -18, 0, -14, -2, 0, 0, 0, 0, 30, 26, 0, -30, 0, 2, 0, -26, 0, -18, 10, 0, -34, 0, 0, 26, 22, 0, 18, 0, -10, 34, 0, 0, 14, 0, 0, -34, 38, 2, -6, 0, 30, 0, 34, 0, 0, -14, 42, 38, 0, 0, 0, 0, 0, 0, 0, -10, -22, 0, -42

Offset: 1

Wolfdieter Lang, Dec 11 2016

This sequence gives also the p-defects for the congruences y^2 == x^3 - x (mod p), y^2 == x^3 - 11*x - 14 (mod p) and y^2 == x^3 - 11*x + 14 (mod p). See the Cremona link, Table 1, N = 32. - Wolfdieter Lang, Dec 22 2016
This elliptic curve y^2 = x^3 + 4*x appears as strong Weil curve for the weight 2 newform (eta(4*tau)*eta(8*tau))^2 of level N=32, with Dedekind's eta function. See the Martin-Ono link, Theorem 2, p. 3173, the row with Conductor 32. See also A002171 for the expansion of this newform in powers of q^4 (but with different offset). The also Nr. 49 of the Martin Table 1.
From this L-series of this elliptic curve one has:
a(n) = 0 if prime(n) == 2 or 3 (mod 4). (see the conjecture by Robert Israel, Sep 28 2016 in A276730).
If prime(n) == 1 (mod 4) = A002144(m) (for a unique m = m(n)) then prime(n) = A(m)^2 + B(m)^2 with the odd A(m) = A002972(m) and the even B(m) = 2*A002973(m). It turns out that 4*A002144(m) - a(m^2) = (2*B(m))^2 for m=m(n), and the sign s(m) of a(m) is + if A(m) + B(m) == 1 (mod 4) and - if A(m) + B(m) == 3 (mod 4). For the primes == 1 (mod 4) leading to sign + or - see A279392 or A279393, respectively. One has thus s(m) = (-1)^((A(m)-1)/2 + B(m)/2). See the Martin-Ono formula for a_{32}(p) in Theorem 3, p. 3175. This leads to the a(n) formula given below.

			a(1) = 0  because prime(1) = 2 == 2 (mod 4).
a(2) = 0 because prime(2) = 3 == 3 (mod 4).
a(3) = -2 because prime(3) = 5 = A002144(1) = A002972(1)^2 + (2*A002973(1))^2 = 1^2 + 2^2. Hence 2*A(1) = 2*A002972(1) = 2, and the sign s(1) = - because A(1) + B(1) = 1 + 2*1 = 3 == 3 (mod 4).
a(6) = +6 because prime(6) = 13 = A002144(2) = A(2)^2 + B(2)^2 = 3^2 + (2*1)^2. Hence 2*A(2) = 6 and the sign is + because A(2) + B(2) = 3 + 2 = 5 == 1 (mod 4).

Seiichi Manyama, Table of n, a(n) for n = 1..10000
George E. Andrews, Number Theory, see S(1) expression p. 135.
J. E. Cremona, Algorithms for Modular Elliptic Curves.
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Yves Martin and Ken Ono, Eta-Quotients and Elliptic Curves, Proc. Amer. Math. Soc. 125, No 11 (1997), 3169-3176.

Cf. A000040, A002144, A002171, A002972, A002973, A279392, A279393.

a(n) =  my(p=prime(n)); -sum(k=1, p-3, kronecker(k*(k+1)*(k+2), p)); \\ Michel Marcus, Nov 06 2020

a(n) = 0 if prime(n) == 2 or 3 (mod 4) (this is conjecture II from above).
a(n) = s(m)*2*A(m) if prime(n) = A002144(m), with A(m) = A002972(m) and the sign s(m) = (-1)^((A(m)-1)/2 + B(m)/2).
a(n) = - Sum_{k=1..p-3} ((k*(k+1)*(k+2))/p) where (x/y) is the Kronecker symbol. - Michel Marcus, Nov 06 2020

See the Jon Perry comment on A005098. The (even) promic number a(n)*(a(n)+1) (see A002378) when subtracted from A005098(n) (the n-th value k such that 4*k+1 is prime) leaves a square, namely A002973(n)^2. E.g., n=4: 7 - 2*3 = 1^1. n=5: 9 - 0 = 3^2.

2*a(n)+1 = A002972(n), n>=1. E.g., n=4: 2*2+1=5.

The primes from A002144 (1 (mod 4) primes) have the property A002144(n) = A002972(n)^2 + (2*A002973(n))^2 = A(n)^2 + B(n)^2 with odd A(n) and even B(n). A bisection of A002144 is given depending on A(n) + B(n) == 1 (mod 4) (part I) or A(n) + B(n) == 3 (mod 4) (part II). The present sequence gives part I of this bisection. The other part II is given in A279393.

This bisection appears in the formula for the p-defects of the congruence y^2 == x^3 + 4*x (mod p) for primes p == 1 (mod 4). See A278720 where for nonvanishing entries the sign is conjectured to be + for these part I primes, and it is - for the part II primes from A279393.

cp's OEIS Frontend

A208295 a(n)*(a(n)+1) + A002973(n)^2 = A005098(n), n>=1.

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A279392 Bisection of primes congruent to 1 modulo 4 (A002144), depending on the corresponding sum of the A002972 and 2*A002973 entries being congruent to 1 modulo 4 or not. Here we give the first case.

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A279393 Bisection of primes congruent to 1 modulo 4 (A002144), depending on the corresponding sum of the A002972 and 2*A002973 entries being congruent to 1 modulo 4 or not. Here we give the second case.

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A002144 Pythagorean primes: primes of the form 4*k + 1.

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A002330 Value of y in the solution to p = x^2 + y^2, x <= y, with prime p = A002313(n).

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A005098 Numbers k such that 4k + 1 is prime.

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A002331 Values of x in the solution to p = x^2 + y^2, x <= y, with prime p = A002313(n).

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